Foundations of Image Science
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"...an impressive tour-de-force textbook of imaging systemsthat deserves a place on the bookshelves of many radiologists andstudents." (Yale Journal of Biology and Medicine, July 2005) "...a worthwhile addition to the armamentarium of any seriousresearcher in image science and will be an opt-quoted reference formany years to come." (Journal of Electrical Imaging,April-June 2005) In an article whether educational programs for imagingphysicists should emphasize science of imaging rather than thetechnology of imaging, "The new textbook...does appear to beoutstanding. It contains over 1500 pages of text, with probablyabout as many equations..." (Medical Physics, October2004) "Foundation of Image Science is comprehensive and themathematics is rigorous and ubiquitous." (E-Streams,Vol. 7, No. 5) "Containing a clear, detailed and general mathematicaldescription of image formation, representation, and qualityassessment, this book will be of great interest to researches andgraduate students...highly recommended." (MedicalPhysics)More details
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Content
1.1 LINEAR VECTOR SPACES.
1.1.1 Vector addition and scalar multiplication.
1.1.2 Metric spaces and norms.
1.1.3 Sequences of vectors and complete metric spaces.
1.1.4 Scalar products and Hilbert space.
1.1.5 Basis vectors.
1.1.6 Continuous bases.
1.2 TYPES OF OPERATORS.
1.2.1 Functions and functionals.
1.2.2 Integral transforms.
1.2.3 Matrix operators.
1.2.4 Continuous-to-discrete mappings.
1.2.5 Differential operators.
1.3 HILBERT-SPACE OPERATORS.
1.3.1 Range and domain.
1.3.2 Linearity, boundedness and continuity.
1.3.3 Compactness.
1.3.4 Inverse operators.
1.3.5 Adjoint operators.
1.3.6 Projection operators.
1.3.7 Outer products.
1.4 EIGENANALYSIS.
1.4.1 Eigenvectors and eigenvalue spectra.
1.4.2 Similarity transformations.
1.4.3 Eigenanalysis infinite-dimensional spaces.
1.4.4 Eigenanalysis of Hermitian operators.
1.4.5 Diago nalization of a Hermitian operator.
1.4.6 Simultaneo us diagonalization of Hermitian matrices.
1.5 SINGULAR-VALUE DECOMPOSITION.
1.5.1 Definition and properties.
1.5.2 Subspaces.
1.5.3 SVD representation of vectors and operators.
1.6 MOORE-PENROSE PSEUDOINVERSE.
1.6.1 Penrose equations.
1.6.2 Pseudoinverses and SVD.
1.6.3 Properties of the pseudoinverse.
1.6.4 Pseudoinverses and projection operators.
1.7 PSEUDOINVERSES AND LINEAR EQUATIONS.
1.7.1 Nature of solutions of linear equations.
1.7.2 Existence and uniqueness of exact solutions.
1.7.3 Explicit solutions for consistent data.
1.7.4 Least-squares solutions.
1.7.5 Minimum-norm solutions.
1.7.6 Iterative calculation of pseudoinverse solution.
1.8 REPRODUCING-KERNEL HILBERT SPACES.
1.8.1 Positive-definite Hermitian operators.
1.8.2 Nonnegative-definite Hermitian operators.
2. THE DIRAC DELTA AND OTHER GENERALIZED FUNCTIONS.
2.1 THEORY OF DISTRIBUTIONS.
2.1.1 Basic concepts.
2.1.2 Well-behaved functions.
2.1.3 Approximation of other functions.
2.1.4 Formal definition of distributions.
2.1.5 Properties of distributions.
2.1.6 Tempered distributions.
2.2 ONE-DIMENSIONAL DELTA FUNCTION.
2.2.1 Intuitive definition and elementary properties.
2.2.2 Limiting representations.
2.2.3 Distributional approach.
2.2.4 Derivatives of delta functions.
2.2.5 A synthesis.
2.2.6 Delta functions as basis vectors.
2.3 OTHER GENERALIZED FUNCTIONS IN 1D.
2.3.1 Generalized functions as limits.
2.3.2 Generalized functions related to the delta function.
2.3.3 Other point singularities.
2.4 MULTIDIMENSIONAL DELTA FUNCTIONS.
2.4.1 Multidimensional distributions.
2.4.2 Multidimensional delta functions.
2.4.3 Delta functions in polar coordinates.
2.4.4 Line masses and plane masses.
2.4.5 Multidimensional derivatives of delta functions.
2.4.6 Other point singularities.
2.4.7 Angular delta functions.
3. FOURIER ANALYSIS.
3.1 SINES, COSINES AND COMPLEX EXPONENTIALS.
3.1.1 Orthogonality on a finite interval.
3.1.2 Complex exponentials.
3.1.3 Orthogonality on the infinite interval.
3.1.4 Discrete orthogonality.
3.1.5 The view from the complex plane.
3.2 FOURIER SERIES.
3.2.1 Basic concepts.
3.2.2 Convergence of the Fourier series.
3.2.3 Properties of the Fourier coefficients.
3.3 1D FOURIER TRANSFORM.
3.3.1 Basic concepts.
3.3.2 Convergence issues.
3.3.3 Unitarity of the Fourier operator.
3.3.4 Fourier transforms of generalized functions.
3.3.5 Properties of the 1D Fourier transform.
3.3.6 Convolution and correlation.
3.3.7 Fourier transforms of some special functions.
3.3.8 Relation between Fourier series and Fourier transforms.
3.3.9 Analyticity of Fourier transforms.
3.3.10 Related transforms.
3.4 MULTIDIMENSIONAL FOURIER TRANSFORMS.
3.4.1 Basis functions.
3.4.2 Definitions and elementary properties.
3.4.3 Multidimensional convolution and correlation.
3.4.4 Rotationally symmetric functions.
3.4.5 Some special functions and their transforms.
3.4.6 Multidimensional periodicity.
3.5 SAMPLING THEORY.
3.5.1 Bandlimited functions.
3.5.2 Reconstruction of a bandlimited function from uniform samples.
3.5.3 Aliasing.
3.5.4 Sampling in frequency space.
3.5.5 Multidimensional sampling.
3.5.6 Sampling with a finite aperture.
3.6 DISCRETE FOURIER TRANSFORM.
3.6.1 Motivation and definitions.
3.6.2 Basic properties of the DFT.
3.6.3 Relation between discrete and continuous Fourier transforms.
3.6.4 Discrete-space Fourier Transform.
3.6.5 Fast Fourier Transform.
3.6.6 Multidimensional DFTs.
4. SERIES EXPANSIONS AND INTEGRAL TRANSFORMS.
4.1 EXPANSIONS IN ORTHOGONAL FUNCTIONS.
4.1.1 Basic concepts.
4.1.2 Orthogonal polynomials.
4.1.3 Sturm-Liouville theory.
4.1.4 Classical orthogonal polynomials and related functions.
4.1.5 Prolate spheroidal wavefunctions.
4.2 CLASSICAL INTEGRAL TRANSFORMS.
4.2.2 Mellin transform.
4.2.3 Z transform.
4.2.4 Hilbert transform.
4.2.5 Higher-order Hankel transforms.
4.3 FRESNEL INTEGRALS AND TRANSFORMS.
4.3.1 Fresnel integrals.
4.3.2 Fresnel transforms.
4.3.3 Chirps and Fourier transforms.
4.4 RADON TRANSFORM.
4.4.1 2D Radon transform and its adjoint.
4.4.2 Central-slice theorem.
4.4.3 Filtered backprojection.
4.4.4 Unfiltered backprojection.
4.4.5 Radon transform in higher dimensions.
4.4.6 Radon transform in signal processing.
5. MIXED REPRESENTATIONS.
5.1 LOCAL SPECTRAL ANALYSIS.
5.1.1 Local Fourier transforms.
5.1.2 Uncertainty.
5.1.3 Local frequency.
5.1.4 Gabor's signal expansion.
5.2 BILINEAR TRANSFORMS.
5.2.1 Wigner distribution function.
5.2.2 Ambiguity functions.
5.2.3 Fractional Fourier transforms.
5.3 WAVELETS.
5.3.1 Mother wavelets and scaling functions.
5.3.2 Continuous wavelet transform.
5.3.3 Discrete wavelet transform.
5.3.4 Multiresolution analysis.
6. GROUP THEORY.
6.1 BASIC CONCEPTS.
6.1.1 Definition of a group.
6.1.2 Group multiplication tables.
6.1.3 Isomorphism and homomorphism.
6.2 SUBGROUPS AND CLASSES.
6.2.1 Definitions.
6.2.2 Examples.
6.3 GROUP REPRESENTATIONS.
6.3.1 Matrices that obey the multiplication table.
6.3.2 Irreducible representations.
6.3.3 Characters.
6.3.4 Unitary irreducible representations and orthogonality.
properties.
6.4 SOME FINITE GROUPS.
6.4.1 Cyclic groups.
6.4.2 Dihedral groups.
6.5 CONTINUOUS GROUPS.
6.5.1 Basic properties.
6.5.2 Linear, orthogonal and unitary groups.
6.5.3 Abelian and non-Abelian Lie groups.
6.6 GROUPS OF OPERATORS ON A HILBERT SPACE.
6.6.1 Geometrical transformations of functions.
6.6.2 Invariant subspaces.
6.6.3 Irreducible subspaces.
6.6.4 Orthogonality of basis functions.
6.7 QUANTUM MECHANICS AND IMAGE SCIENCE.
6.7.1 A smattering of quantum mechanics.
6.7.2 Connection with image science.
6.7.3 Symmetry group of the Hamiltonian.
6.7.4 Symmetry and degeneracy.
6.7.5 Reducibility and accidental degeneracy.
6.7.6 Parity.
6.7.7 Rotational symmetry in three dimensions.
6.8 FUNCTIONS AND TRANSFORMS ON GROUPS.
6.8.1 Functions on a finite group.
6.8.2 Extension to infinite groups.
6.8.3 Convolutions on groups.
6.8.4 Fourier transforms on groups.
6.8.5 Wavelets revisited.
7. DETERMINISTIC DESCRIPTIONS OF IMAGING SYSTEMS.
7.1 OBJECTS AND IMAGES.
7.1.1 Objects and images as functions.
7.1.2 Objects and images as infinite-dimensional vectors.
7.1.3 Objects and images as finite-dimensional vectors.
7.1.4 Representation accuracy.
7.1.5 Uniform translates.
7.1.6 Other representations.
7.2 LINEAR CONTINUOUS-TO-CONTINUOUS SYSTEMS.
7.2.1 General shift-variant systems.
7.2.2 Adjoint operators and SVD.
7.2.3 Shift-invariant systems.
7.2.4 Eigenanalysis of LSIV systems.
7.2.5 Singular-value decomposition of LSIV systems.
7.2.6 Transfer functions.
7.2.7 Magnifiers.
7.2.8 Approximately shift-invariant systems.
7.2.9 Rotationally symmetric systems.
7.2.10 Axial systems.
7.3 LINEAR CONTINUOUS-TO-DISCRETE SYSTEMS.
7.3.1 System operator.
7.3.2 Adjoint operator and SVD.
7.3.3 Fourier description.
7.3.4 Sampled LSIV systems.
7.3.5 Mixed CC-CD systems.
7.3.6 Discrete-to-continuous systems.
7.4 LINEAR DISCRETE-TO-DISCRETE SYSTEMS.
7.4.1 System matrix.
7.4.2 Adjoint operator and SVD.
7.4.3 Image errors.
7.4.4 Discrete representations of shift-invariant system.
7.5 NONLINEAR SYSTEMS.
7.5.1 Point nonlinearities.
7.5.2 Nonlocal nonlinearities.
7.5.3 Object-dependent system operators.
7.5.4 Postdetection nonlinear operations.
8. STOCHASTIC DESCRIPTIONS OF OBJECTS AND IMAGES.
8.1 RANDOM VECTORS.
8.1.1 Basic concepts.
8.1.2 Expectations.
8.1.3 Covariance and correlation matrices.
8.1.4 Characteristic functions.
8.1.5 Transformations of random vectors.
8.1.6 Eigenanalysis of covariance matrices.
8.2 RANDOM PROCESSES.
8.2.1 Definitions and basic concepts.
8.2.2 Averages of random processes.
8.2.3 Characteristic functionals.
8.2.4 Correlation analysis.
8.2.5 Spectral analysis.
8.2.6 Linear filtering of random processes.
8.2.7 Eigenanalysis of the autocorrelation operator.
8.2.8 Discrete random processes.
8.3 NORMAL RANDOM VECTORS AND PROCESSES.
8.3.1 Probability density functions.
8.3.2 The characteristic function.
8.3.3 Marginal densities and linear transformations.
8.3.4 Central-limit theorem.
8.3.5 Normal random processes.
8.3.6 Complex Gaussian random fields.
8.4 STOCHASTIC MODELS FOR OBJECTS.
8.4.1 Probability density functions in Hilbert space.
8.4.2 Multipoint densities.
8.4.3 Normal models.
8.4.4 Texture models.
8.4.5 Signals and backgrounds.
8.5 STOCHASTIC MODELS FOR IMAGES.
8.5.1 Linear systems.
8.5.2 Conditional statistics for a single object.
8.5.3 Effects of object randomness.
8.5.4 Signals and backgrounds in image space.
9. DIFFRACTION THEORY AND IMAGING.
9.1 WAVE EQUATIONS.
9.1.1 Maxwell's equations.
9.1.2 Maxwell's equations in the Fourier domain.
9.1.3 Material media.
9.1.4 Time-dependent wave equations.
9.1.5 Time-independent wave equations.
9.2 PLANE WAVES AND SPHERICAL WAVES.
9.2.1 Plane waves.
9.2.2 Spherical waves.
9.3 GREEN'S FUNCTIONS.
9.3.1 Differential equations for the Green's functions.
9.3.2 Free-space time-dependent Green's function.
9.3.3 Free-space GF for the Helmholtz and Poisson equations.
9.3.4 Defined-source problems.
9.3.5 Boundary-value problems.
9.4 DIFFRACTION BY A PLANAR APERTURE.
9.4.1 The surface at infinity.
9.4.2 Kirchhoff boundary conditions.
9.4.3 Application of Green's theorem.
9.4.4 Diffraction as a 2D linear filter.
9.4.5 Some useful approximations.
9.4.6 Fresnel dffraction.
9.4.7 Fraunhofer diffraction.
9.5 DIFFRACTION IN THE FREQUENCY DOMAIN.
9.5.1 Angular spectrum.
9.5.2 Fresnel and Fraunhofer approximations.
9.5.3 Beams.
9.5.4 Reection and refraction of light.
9.6 IMAGING OF POINT OBJECTS.
9.6.1 The ideal thin lens.
9.6.2 Imaging of a monochromatic point source.
9.6.3 Transmittance of an aberrated lens.
9.6.4 Rotationally symmetric lenses.
9.6.5 Field curvature and distortion.
9.6.6 Probing the pupil.
9.6.7 Interpretation of the other Seidel aberrations.
9.7 IMAGING OF EXTENDED PLANAR OBJECTS.
9.7.1 Monochromatic objects and a simple lens.
9.7.2 A more complicated imaging system.
9.7.3 Random fields and coherence.
9.7.4 Quasimonochromatic imaging.
9.7.5 Spatially incoherent, quasimonochromatic imaging.
9.7.6 Polychromatic, incoherent imaging.
9.7.7 Partially coherent imaging.
9.8 VOLUME DIFFRACTION AND 3D IMAGING.
9.8.1 The Born approximation.
9.8.2 The Rytov approximation.
9.8.3 Fraunhofer diffraction from volume objects.
9.8.4 Coherent 3D imaging.
10. ENERGY TRANSPORT AND PHOTONS.
10.1 ELECTROMAGNETIC ENERGY FLOW AND DETECTION.
10.1.1 Energy ow in classical electrodynamics.
10.1.2 Plane waves.
10.1.3 Photons.
10.1.4 Physics of photodetection.
10.1.5 What do real detectors detect?.
10.2 RADIOMETRIC QUANTITIES AND UNITS.
10.2.1 Self-luminous surface objects.
10.2.2 Self-luminous volume objects.
10.2.3 Surface reection and scattering.
10.2.4 Transmissive objects.
10.2.5 Cross sections.
10.2.6 Distribution function.
10.2.7 Radiance in physical optics and quantum optics.
10.3 THE BOLTZMANN TRANSPORT EQUATION.
10.3.1 Derivation of the Boltzmann equation.
10.3.2 Steady-state solutions in non-absorbing media.
10.3.3 Steady-state solutions in absorbing media.
10.3.4 Scattering effects.
10.3.5 Spherical harmonics.
10.3.6 Elastic scattering and diffusion.
10.3.7 Inelastic (Compton) scattering.
10.4 TRANSPORT THEORY AND IMAGING.
10.4.1 The general imaging equation.
10.4.2 Pinhole imaging.
10.4.3 Optical imaging of planar objects.
10.4.4 Adjoint methods.
10.4.5 Monte Carlo methods.
11. POISSON STATISTICS AND PHOTON COUNTING.
11.1 POISSON RANDOM VARIABLES.
11.1.1 Poisson and independence.
11.1.2 Poisson and rarity.
11.1.3 Binomial selection of a Poisson.
11.1.4 Doubly stochastic Poisson random variables.
11.2 POISSON RANDOM VECTORS.
11.2.1 Multivariate Poisson statistics.
11.2.2 Doubly stochastic multivariate statistics.
11.3 RANDOM POINT PROCESSES.
11.3.1 Temporal point processes.
11.3.2 Spatial point processes.
11.3.3 Mean and autocorrelation of point processes.
11.3.4 Relation between Poisson random vectors and processes.
11.3.5 Karhunen-Loéve analysis of Poisson processes.
11.3.6 Doubly stochastic spatial Poisson random processes.
11.3.7 Doubly stochastic temporal Poisson random processes.
11.3.8 Point processes in other domains.
11.3.9 Filtered point processes.
11.3.10 Characteristic functionals of filtered point processes.
11.3.11 Spectral properties of point processes.
11.4 RANDOM AMPLIFICATION.
11.4.1 Random amplification in single-element detectors.
11.4.2 Random amplification and generating functions.
11.4.3 Random amplification of point processes.
11.4.4 Spectral analysis.
11.4.5 Random amplification in arrays.
11.5 QUANTUM MECHANICS OF PHOTON COUNTING.
11.5.1 Coherent states.
11.5.2 Density operators.
11.5.3 Counting statistics.
12. NOISE IN DETECTORS.
12.1 PHOTON NOISE AND SHOT NOISE IN PHOTODIODES.
12.1.1 Vacuum photodiodes.
12.1.2 Basics of semiconductor detectors.
12.1.3 Shot noise in semiconductor photodiodes.
12.2 OTHER NOISE MECHANISMS.
12.2.1 Thermal noise.
12.2.2 Generation-recombination noise.
12.2.3 1/f noise.
12.2.4 Noise in gated integrators.
12.2.5 Arrays of noisy photodetectors.
12.3 X-RAY AND GAMMA-RAY DETECTORS.
12.3.1 Interaction mechanisms.
12.3.2 Photon-counting semiconductor detectors.
12.3.3 Semiconductor detector arrays.
12.3.4 Position and energy estimation with semiconductor detectors.
12.3.5 Scintillation cameras.
12.3.6 Position and energy estimation with scintillation cameras.
12.3.7 Imaging characteristics of photon-counting detectors.
12.3.8 Integrating detectors.
12.3.9 K x rays and Compton scattering.
13. STATISTICAL DECISION THEORY.
13.1 BASIC CONCEPTS.
13.1.1 Kinds of decisions.
13.2 CLASSIFICATION TASKS.
13.2.1 Partitioning the data space.
13.2.2 Binary decision outcomes.
13.2.3 The ROC curve.
13.2.4 Performance measures for binary tasks.
13.2.5 Computation of AUC.
13.2.6 The likelihood ratio and the ideal observer.
13.2.7 Statistical properties of the likelihood ratio.
13.2.8 Ideal observer with Gaussian statistics.
13.2.9 Ideal observer with non-Gaussian statistics.
13.2.10 Signal variability and the ideal observer.
13.2.11 Background variability and the ideal observer.
13.2.12 The optimal linear discriminant.
13.2.13 Detectability in continuous data.
13.3 ESTIMATION THEORY.
13.3.1 Basic concepts.
13.3.2 MSE in digital imaging.
13.3.3 Bayesian estimation.
13.3.4 Maximum-likelihood estimation.
13.3.5 Likelihood and Fisher information.
13.3.6 Properties of ML estimators.
13.3.7 Other classical estimators.
13.3.8 Nuisance parameters.
13.3.9 Hybrid detection/estimation tasks.
14. IMAGE QUALITY.
14.1 SURVEY OF APPROACHES.
14.1.1 Subjective assessment.
14.1.2 Fidelity measures.
14.1.3 JND models.
14.1.4 Information-theoretic assessment.
14.1.5 Objective assessment of image quality.
14.2 HUMAN OBSERVERS AND CLASSIFICATION TASKS.
14.2.1 Methods for investigating the visual system.
14.2.2 Modified ideal-observer models.
14.2.3 Psychophysical methods for image evaluation.
14.2.4 Estimation of figures of merit.
14.3 MODEL OBSERVERS AND CLASSIFICATION TASKS.
14.3.1 General considerations.
14.3.2 Linear observers.
14.3.3 Ideal observers.
14.3.4 Estimation tasks.
14.4 ESTIMATION TASKS.
14.4.1 Performance metrics.
14.4.2 Estimation of linear parameters.
14.4.3 Estimation of nonlinear parameters.
14.4.4 System optimization for estimation tasks.
14.5 SOURCES OF IMAGES.
14.5.1 Deterministic simulation of objects.
14.5.2 Stochastic simulation of objects.
14.5.3 Deterministic simulation of image formation.
14.5.4 Stochastic simulation of image formation.
14.4.5 Gold standards.
15. INVERSE PROBLEMS.
15.1 BASIC CONCEPTS.
15.1.1 Classification of inverse problems.
15.1.2 The discretization dilemma.
15.1.3 Estimability.
15.1.4 Positivity.
15.1.5 Choosing the best algorithm.
15.2 LINEAR RECONSTRUCTION OPERATORS.
15.2.1 Matrix operators for estimation of expansion coefficients.
15.2.2 Reconstruction of functions from discrete data.
15.2.3 Reconstruction from Fourier samples.
15.2.4 Discretization of analytic inverses.
15.2.5 More on analytic inverses.
15.2.6 Noise with linear reconstruction operators.
15.3 IMPLICIT ESTIMATES.
15.3.1 Functional minimization.
15.3.2 Data-agreement functionals.
15.3.3 Regularizing functionals.
15.3.4 Effects of positivity.
15.3.5 Reconstruction without discretization.
15.3.6 Resolution and noise in implicit estimates.
15.4 ITERATIVE ALGORITHMS.
15.4.1 Linear iterative algorithms.
15.4.2 Noise propagation in linear algorithms.
15.4.3 Search algorithms for functional minimization.
15.4.4 Nonlinear constraints and fixed-point iterations.
15.4.5 Projections onto convex sets.
15.4.6 The MLEM algorithm.
15.4.7 Noise propagation in nonlinear algorithms.
15.4.8 Stochastic algorithms.
16. PLANAR IMAGING WITH X RAYS AND GAMMA RAYS.
16.1 DIGITAL RADIOGRAPHY.
16.1.1 The source and the object.
16.1.2 X-ray detection.
16.1.3 Scattered radiation.
16.1.4 Deterministic properties of shadow images.
16.1.5 Stochastic properties.
16.1.6 Image quality: Detection tasks.
16.1.7 Image quality: Estimation tasks.
16.2 PLANAR NUCLEAR MEDICINE.
16.2.1 Basic issues.
16.2.2 Image formation.
16.2.3 The detector.
16.2.4 Stochastic properties.
16.2.5 Image quality: Classification tasks.
16.2.6 Image quality: Estimation tasks.
17. EMISSION COMPUTED TOMOGRAPHY.
17.1 FORWARD PROBLEMS.
17.1.1 CD formulations for parallel-beam SPECT.
17.1.2 Equally spaced angles.
17.1.3 Fourier analysis in the CD formulation.
17.1.4 The 2D Radon transform and parallel-beam SPECT.
17.1.5 3D transforms and cone-beam SPECT.
17.1.6 Attenuation.
17.2 INVERSE PROBLEMS.
17.2.1 SVD of the 2D Radon transform.
17.2.2 Inverses and pseudoinverses in 2D.
17.2.3 Inversion of the 3D x-ray transform.
17.2.4 Inversion of attenuated transforms.
17.2.5 Discretization of analytic reconstruction algorithms.
17.2.6 Matrices for iterative methods.
17.3 NOISE AND IMAGE QUALITY.
17.3.1 Noise in the data.
17.3.2 Noise in reconstructed images.
17.3.3 Artifacts.
17.3.4 Image quality.
18. SPECKLE.
18.1 BASIC CONCEPTS.
18.1.1 Elementary statistical considerations.
18.1.2 Speckle in imaging.
18.2 SPECKLE IN A NONIMAGING SYSTEMS.
18.2.1 Description of the ground glass.
18.2.2 Some simplifying assumptions.
18.2.3 Propagation of characteristic functionals.
18.2.4 Central-limit theorem.
18.2.5 Statistics of the irradiance.
18.3 SPECKLE IN AN IMAGING SYSTEM.
18.3.1 The imaging system.
18.3.2 Propagation of characteristic functionals.
18.3.3 Effect of the detector.
18.3.4 Point scatterers.
18.4 NOISE AND IMAGE QUALITY.
18.4.1 Measurement noise.
18.4.2 Random objects.
18.4.3 Task performance.
18.5 POINT-SCATTERINGMODELSANDNON-GAUSSIANSPECKLE.
18.5.1 Object fields and objects.
18.5.2 Image fields.
18.5.3 Univariate statistics of the image field and irradiance.
18.6 COHERENT RANGING.
18.6.1 System configurations.
18.6.2 Deterministic analysis.
18.6.3 Statistical analysis.
18.6.4 Task performance.
19. IMAGING IN FOURIER SPACE.
19.1 FOURIER MODULATORS.
19.1.1 Data acquisition.
19.1.2 Noise.
19.1.3 Reconstruction.
19.1.4 Image quality.
19.2 INTERFEROMETERS.
19.2.1 Young's double-slit experiment.
19.2.2 Visibility estimation.
19.2.3 Michelson stellar interferometer.
19.2.4 Interferometers with multiple telescopes.
EPILOGUE. FRONTIERS IN IMAGE SCIENCE.
Appendix A: MATRIX ALGEBRA.
Appendix B: COMPLEX VARIABLES.
Appendix C: PROBABILITY.
Bibliography.
Index.
Preface
Images are ubiquitous in the modern world. We depend on images for news, communication and entertainment as well as for progress in medicine, science and technology. For better or worse, television images are virtually a sine qua non of modern life. If we become ill, medical images are of primary importance in our care. Satellite images provide us with weather and crop information, and they provide our military commanders with timely and accurate information on troop movements. Biomedical research and materials science could not proceed without microscopic images of many kinds. The petroleum reserves so essential to our economy are usually found through seismic imaging, and enemy submarines are located with sonic imaging. These examples, and many others that readily come to mind, are ample proof of the importance of imaging systems.
While many of the systems listed above involve the latest in high technology, it is not so obvious that there is an underlying intellectual foundation that ties the technologies together and enables systematic design and optimization of diverse imaging systems. A substantial literature exists for many of the subdisciplines of image science, including quantum optics, ray optics, wave propagation, image processing and image understanding, but these topics are typically treated in separate texts without significant overlap. Moreover, the practitioner's goal is to make better images, in some sense, but little attention is paid to the precise meaning of the word "better." In such circumstances, can imaging be called a science?
There are three elements that must be present for a discipline to be called a science. First, the field should have a common language, an agreed-upon set of definitions. Second, the field should have an accepted set of experimental procedures. And finally, the field should have a theory with predictive value. It is the central theme of this book that there is indeed a science of imaging, with a well-defined theoretical and experimental basis. In particular, we believe that image quality can be defined objectively, measured experimentally and predicted and optimized theoretically.
Our goal in writing this book, therefore, is to present a coherent treatment of the mathematical and physical foundations of image science and to bring image evaluation to the forefront of the imaging community's consciousness.
ORGANIZATION OF THE BOOK
There are a number of major themes that weave their way throughout this book, as well as philosophical stances we have taken, so we recommend that the reader begin with the prologue to get an introduction to these themes and our viewpoint. Once this big picture is absorbed, the reader should be ready to choose where to jump into the main text for more detailed reading.
Mathematical Foundations
The first six chapters of this book represent our estimation of the essential mathematical underpinnings of image science. In our view, anyone wishing to do advanced research in this field should be conversant with all of the main topics presented there. The first four chapters are devoted to the important tools of linear algebra, generalized functions, Fourier analysis and other linear transformations. Chapter 5 treats a class of mathematical descriptions called mixed representations, that is, descriptions that mix seemingly incompatible variables such as spatial position and spatial frequency. Chapter 6 presents the basic concepts of group theory, the mathematics of symmetry, which will be applied to the description of imaging systems in later chapters.
It was our objective in writing these introductory chapters to present the mathematical foundations of image science at a level that will be accessible to graduate students and well-motivated undergraduates. At the same time, we have attempted to include sufficient advanced material so that the material will be beneficial to established workers in the field. This dual goal requires examining many concepts at different levels of sophistication. We have attempted to do this by providing both elementary explanations of the key points and more detailed mathematical treatments. The reader will find that the level of mathematical rigor is not uniform throughout these chapters or even within a particular chapter. We hope that this approach allows each reader to extract from the book insights appropriate to his or her individual interests and mathematical preparation.
Image Formation: Models and Mechanisms
A quick perusal of the of Contents will reveal that a significant portion of the book is devoted to the subject of image formation. We have strived to present a comprehensive and unified treatment of the mathematical and statistical principles of imaging. We hope this serves the image-science community by giving a common language and framework to our many disciplines. Additionally, a thorough understanding of the image-formation process is a prerequisite for the image-evaluation methodology we advocate.
The deterministic analysis of imaging systems begins in Chap. 7, where we present a wide variety of mathematical descriptions of objects and images and mappings from object to image. We argue in Chap. 7 (and briefly also in the Prologue) that digital imaging systems are best described as a mapping from a function to a discrete set of numbers, so much of the emphasis in that chapter will be on such mappings. More conventional mappings such as convolutions are, however, also treated in a unified way. An important tool in Chap. 7 is singular-value decomposition, which is introduced mathematically in Chap. 1.
The deterministic mappings are not a complete description of image formation. Repeated images of a single object will not be identical because of electronic noise in detectors and amplifiers, as well as photon noise, which arises from the discrete nature of photoelectric interactions. In addition, the object itself can often be usefully regarded as random. Object statistics are important in pattern recognition, image reconstruction and evaluation of image quality. Chapter 8 provides a general mathematical framework for the description of random vectors and processes. Particular emphasis is given to Gaussian random vectors and processes, which often arise as the result of the central-limit theorem.
The next two chapters go more deeply into specific mechanisms of image formation. Chapter 9 develops the theory of wave propagation from first principles and treats diffraction and imaging with waves within this framework. Though the objective of the discussion is to develop deterministic models of wave-optical imaging systems, we cannot avoid discussing random processes when we consider the coherence properties of wave fields, so an understanding of the basics of random processes, as presented in Chap. 8, is needed for a full understanding of Chap. 9. The reader with previous exposure to such topics as autocorrelation functions and complex Gaussian random fields can, however, skip Chap. 8 and move directly to Chap. 9.
Chapter 10 is ostensibly devoted to radiometry and radiative transport, but actually it covers a wide variety of topics ranging from quantum electrodynamics to tomographic imaging. A key mathematical tool developed in that chapter is the Boltzmann equation, a general integro-differential equation that is capable of describing virtually all imaging systems in which interference and diffraction play no significant role. The Boltzmann equation describes a distribution function that can loosely be interpreted as a density of photons in phase space, so it is necessary to discuss in that chapter just what we mean by the ubiquitous word photon.
In Chap. 10 we discuss only the mean photon density or the mean rate of photoelectric interactions, but in Chap. 11 we begin to discuss fluctuations about these means. In particular, we present there an extensive discussion of the Poisson probability law and its application to simple counting detectors and imaging arrays. Included is a discussion of photon counting from a quantum-mechanical perspective. Many of the basic principles of random vectors and processes enunciated in Chap. 8 are used in Chap. 11.
Chapter 12 goes into more detail on noise mechanisms in various detectors of electromagnetic radiation. The implications of Poisson statistics are discussed in practical terms, and a number of noise mechanisms that are not well described by the Poisson distribution are introduced. A long section is devoted to x-ray and gamma-ray detectors, not only because of their practical importance in medical imaging, but also because they illustrate some important aspects of the theory developed in Chap. 11.
Inferences from Images
With the background developed in Chaps. 1-12, we can discuss ways of drawing inferences from image data. The central mathematical tool we need for this purpose is statistical decision theory, introduced in Chap. 13. This theory allows a systematic approach to estimation of numerical parameters from image data as well as classifying the object that produced a given image, and it will form the cornerstone of our treatment of image quality. In accordance with this theory, we shall define image quality in terms of how well an observer can extract some desired information from an image.
Chapter 14 is a nuts-and-bolts guide to objective assessment of image quality for both hardware and software. Particular attention is paid to evaluating the performance of human observers, for whom most images are intended.
Chapter 15 provides a general treatment of inverse problems or image reconstruction, defined as inferring properties of an object from data that do not initially...
